4 Answers
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The big difference is that cumulative is far more common than accumulative.

At the level of actual meaning, to the extent that accumulative is used at all, it tends to refer to someone/something doing the accumulating. By contrast, cumulative is more associated with that which is accumulated.

If the sense intended is acquisitive, just use that word. In all other cases, use cumulative.

Accumulative lays more emphasis on the process of accumulating, and cumulative on the result. [Parallel to FumbleFingers' agent vs. substance distinction but perhaps an iota more transparent.]

Note that accumulative derives from ad- ("in addition") plus cumulare, "heap up", from cumulus, "heap". As noted in wiki.answers, prefix ac "replaces ad as a prefix when used in words where ad doesn't conjoin well with the root word e.g. acclaim, account, accomplice." (Ac- and ad- also are briefly mentioned in a useful chart of Roots, Prefixes, Suffixes pdf file.) The 16th-century reasons for adding on adding on to adding on may be undiscoverable; of course the two words express minutely different meanings, but whether they express a difference worth expressing is far less clear.

In modern usage, there are numerous phrases (cumulative distribution in particular) where use of accumulative would appear illiterate; typically, one can avoid accumulative altogether. Of course other forms of the word cannot be avoided, as in accumulate wealth and accumulator register.

(Editing Note: I removed an either ... or construction from the previous paragraph, in response to FumbleFingers' comment, based on my misunderstanding of "In all other cases, use cumulative".)

Accumulative means having the characteristic of tending to accumulate.

Cumulative means having built up or accumulated over time. E.g. something has a cumulative effect, or is a cumulative result of something. It is more specific in meaning than 'accumulative', in that it is about a definite build up of one thing after another, as opposed to a general tendency to gather up.

There appears to be a tendency for people to use 'accumulative' incorrectly when they in fact mean 'a cumulative...'.

Without using forms of the words to define them (a linguistic no-no), "cumulative" increasing or increased in quantity, degree, or force by successive additions and "accumulative" means gathering or growing by gradual increases.